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In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on $\mathbb{R}^N$: $u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$. It proves that when the growth exponent $p$ of the nonlinearity $f(u) $ is up to the supercritical range: $ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:$ u_{tt}+Δ^2 u+(-Δ)^{α} u_{t}-Δ f(u) = g(x)$, with $α∈ (0, 1)$. For general bounded domain $Ω\subset \mathbb{R}^N (N≥1)$, we show that there exists a critical exponent $p_α\equiv\frac{N+2(2α-1)}{(N-2)^+}$ depending on the dissipative index *α* such that when the growth *p* of the nonlinearity *f*(*u*) is up to the range: $1≤p <p_α$, (ⅰ) the weak solutions of the equations are of additionally global smoothness when $t>0$; (ⅱ) the related dynamical system possesses a global attractor $\mathcal{A}_α$ and an exponential attractor $\mathcal{A}^α_{exp}$ in natural energy space for each $α∈ (0, 1)$, respectively; (ⅲ) the family of global attractors $\{\mathcal{A}_α\}$ is upper semicontinuous at each point $α_0∈ (0,1] $, i.e., for any neighborhood *U* of $\mathcal{A}_{α_0}, \mathcal{A}_α\subset U$ when $|α-α_0|\ll 1$. These results extend those for structural damping case: $α∈ [1, 2)$ in [

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